Abstract

This paper is an elementary introduction to some new asymptotic methods for the search for the solitary solutions of nonlinear differential equations, nonlinear differential-difference equations, and nonlinear fractional differential equations. Particular attention is paid throughout the paper to giving an intuitive grasp for the variational approach, the Hamiltonian approach, the variational iteration method, the homotopy perturbation method, the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform, and ancient Chinese mathematics. Hamilton principle and variational principles are also emphasized. The reviewed asymptotic methods are easy to be followed for various applications. Some ideas on this paper are first appeared.

1. Introduction

Soliton was first discovered in 1834 by Russell [1], who observed that a canal boat stopping suddenly gave rise to a solitary wave which traveled down the canal for several miles, without breaking up or losing strength. Russell named this phenomenon the “soliton.”

In a highly informative as well as entertaining article [1], Russell gave an engaging historical account of the important scientific observation:

I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.

His ideas did not earn attention until 1965 when Zabusky and Kruskal [2] began to use a finite difference approach to the study of KdV equation, which was obtained by Korteweg and de Vires [3]. Various analytical methods also led to a complete understanding of solitons, especially the inverse scattering transform proposed by Gardner et al. [4] in 1967. The significance of Russell’s discovery was then fully appreciated. It was discovered that many phenomena in physics, electronics, and biology can be described by a mathematical and physical theory of “soliton.” For a historical account of the scientific development of solitons, the reader is referred to the “Encyclopedia of Complexity and Systems Science,” especially [5, 6]. Some analytical methods leading to our present state of the art are available in several review articles [7–9].

A soliton is a special traveling wave that after a collision with another soliton eventually emerges unscathed. Solitons are solutions of partial differential equations that model phenomena like water waves or waves along a weakly anharmonic mass-spring chain. A soliton is a bell-like solution as illustrated in Figure 1.

Figure 1: Bell-like solitary wave.

The soliton can be written in a standard form, which is
where , and is the wave velocity.

It is obvious that
The soliton has exponential tails, which are the basic character of solitary waves. This property allows the exponential function to describe its solution, see Section 5.9 for detailed discussion.

The soliton obeys a superposition-like principle: solitons passing through one another emerge unmodified, see Figure 2.

Figure 2: Collision of two solitary waves.

A compacton is a special solitary traveling wave that, unlike a soliton, does not have exponential tails. A compacton-like solution is a special wave solution which can be expressed by the squares of sinusoidal or cosinoidal functions.

Compactons are special solitons with finite wavelength. It was Rosenau and Hyman [10] who first found compactons in 1993.

3.1. Compacton: An Oscillatory Wave with No Tails

Now consider a modified version of KdV equation in the form
Introducing a complex variable defined as , where is the velocity of traveling wave, integrating once, and we have
where is an integral constant, for solitary solution, and we set .

In case , we have periodic solution: . Periodic solution of nonlinear oscillators can be approximated by sinusoidal function. It helps understanding if an equation can be classified as oscillatory by direct inspection of its terms.

We consider two common-order differential equations whose exact solutions are important for physical understanding:
Both equations have linear terms with constant coefficients.

The crucial difference between these two very simple equations is the sign of the coefficient of in the second term. This determines whether the solutions are exponential or oscillatory. The general solution of (3.4) is
The second equation, (3.5), has a positive coefficient of , and in this case, the general solution reads
This solution describes an oscillation at the angular velocity .

Equation (3.3) behaves sometimes like an oscillator when , that is, has a periodic solution, and we assume that can be expressed in the form
Substituting (3.8) into (3.3) results in
We, therefore, have
Solving the above system, (3.10), yields
We obtain the solution in the form
By a careful inspection, can tend to a very small value or even zero, and as a result, tends to negative infinite, and (3.3) behaves like (3.4) with ; the exponential tails vanish completely at the edge of the bell shape (see Figure 3):
This is a compact wave. Unlike solitons, compacton does not have exponential tails (Figure 3).

Figure 3: Compacton wave without tails.

3.2. A Criterion for Oscillatory Thermopower Waves

Thermal conduction in fuel/Bi2Te3/Al2O3 or fuel/Bi2Te3/terracotta systems always results in strong oscillation of the output signals. A criterion for oscillatory thermopower waves is much needed.

Recently, Walia et al. proposed a theory of thermopower wave oscillations to describe coupled thermal waves in fuel/Bi2Te3/Al2O3 or fuel/Bi2Te3/terracotta systems [11]. The dimensionless governing equations are as follows [11]:
where is nondimensional temperature, is the concentration of the fuel, and are, respectively, parameters related to the properties of the fuel and volumetric heat transfer, and is the ambient temperature.

Ignoring the nonlinear term in (3.14), we would have a wave solution. Changing the parameters in the system will result in strong oscillation [11], and an analytical criterion to predict oscillatory thermopower waves is very useful for design of Bi2Te3 films.

The system, (3.14) and (3.15), is difficult to solve analytically because of strong nonlinearity. In order to obtain a criterion for oscillatory thermopower waves, some necessary approximations are needed. Equation (3.15) is approximately written in the form
in (3.16) is assumed to be a known function; solving in (3.16) results in
The nonlinear term, , in (3.14) is expressed in an approximate form
Equation (3.14) is rewritten in the following equivalent form:
where is defined as
In order to solve thermopower waves, we make a transform
where is wave speed.

By the transform, (3.21), we convert (3.19) into an ordinary differential equation, which is
We rewrite (3.22) in the form
where

Note. Equation (3.23) is exactly equivalent to (3.14). In order to solve (3.23) approximately, we write an iteration formulation
In (3.25), can be considered as a known function of . Equation (3.25) is, therefore, similar to a forced nonlinear oscillator. We search for a periodic solution of (3.25). To this end, we assume that its solution can be expressed in the form
By an analytical method [7], we can obtain the following approximate frequency:
The assumption, (3.26), follows , that is,

This is a criterion for oscillatory thermopower waves. When , we can predict thermopower waves without oscillation.

3.3. A Criterion for Gaseous Emission Waves

Lin and Hildemann [12] developed a general mathematical model to predict emissions of volatile organic compounds (VOCs) from hazardous or sanitary landfills. The model includes important mechanisms occurring in unsaturated subsurface landfill environments: biogas flow, leachate flow, diffusion, adsorption, degradation, volatilization, and mass transfer limitations through the top cover. Lin-Hildemann equation for gaseous emission can be expressed as follows [12]:
with the following boundary/initial conditions:
where is the total chemical concentration per unit volume of soil, is the effective emission speed, and is effective diffusion coefficient. The definitions of and are given in [12].

Equation (3.29) is converted into an ordinary differential equation, which is
where is the wave speed.

Considering the boundary condition, (3.31), the solution of (3.34) is
where
By the boundary condition, (3.30), we have
Solving and from (3.36) and (3.37) results in
Considering the initial condition, we have
Equation (3.40) can be written in the form
where , are the bulk (apparent) gas and water velocities, respectively. , are the effective gaseous and aqueous diffusion coefficients in soil, respectively; , are phase-partitioning coefficients of gas and liquid, respectively.

For a wave solution, the initial condition should be expressed in the form of (3.32). If the initial condition cannot be expressed in an exponential function, the criterion for gaseous emission waves becomes invalid. The present criterion can easily be extended to various nonlinear cases.

4. Exact Solutions versus Asymptotic Solutions

There is plainly a tendency in the modern nonlinear science community to obtain exact solutions for nonlinear equations. There are many results on the exact solutions of nonlinear equations where the initial or boundary conditions are not considered. These solutions are called mathematical solutions because the physical constraints on the real-world problem that is being modeled are not accounted for. Our main aim, however, is to find solutions of the underlying problem that satisfy all the initial/boundary conditions that exist. These solutions, naturally, are called the physical solutions of the problem. Consider, for example, the well-known KdV equation
with a solitary solution
Many mathematical solutions for (4.1) could be found that carry no physical meaning (, for instance, is an exact solution of (4.1) that has no physical meaning at all). Other researchers, on the other hand, begin with some very good initial conditions, say
and find that the condition is in fact too good to solve the equation. For a travelling solution, for example, we might guess a solution of the form
where the unknown constant can be determined by substituting (4.4) into (4.1).

An asymptotic approach is, however, to search for an asymptotic solution with physical understanding. If, for example, we feel interest in a solitary solution of (4.1), then we can assume that the solution has the form
or
where , , , , and are unknown constants which can be determined via various methods.

For -solitary solutions, we can assume that the solution has the following form:
or
where and are unknown constants to be further determined.

For a two-wave solution, we can assume that
or
where .

Some asymptotic methods are easy and accessible to all nonmathematicians using only pencil and paper. Consider a nonlinear differential equation for corneal shape [13]
with boundary conditions and .

Hereby we suggest a Taylor series method to find an asymptotic solution [14].

We rewrite (4.11) in the form
Incorporating the boundary condition, , we have
Differentiating (4.12) with respect to results in
This yields
Proceeding a similar way as above, we have
Applying the Taylor series, we obtain
or
Incorporating the boundary condition, , yields
or
From (4.20), can be solved, which reads [14]
To compare with Okrasiński and Płociniczak’s result, setting , we have
which is very close to Okrasiński and Płociniczak’s result [13].

The accuracy can be further improved if the solution procedure continues.

Comparing the Okrasiński and Płociniczak’s method with our pencil-and-paper method, we conclude that the solution process is accessible to nonmathematicians to solve any nonlinear two-point boundary problems.

5. Asymptotic Methods for Solitary Solutions

The investigation of soliton solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. There are many analytical approaches to the search for solitary solutions, see [5, 6]. Among various methods, the homotopy perturbation method, the variational iteration method, the exp-function method, and the variational approach have been worked out over a number of years by numerous authors, and they have matured into relatively fledged analytical methods for nonlinear equations thanks to the efforts of many researchers.

5.1. Soliton Perturbation

We consider the following perturbed nonlinear evolution equation [5, 15]:
When , we have an unperturbed equation
which is assumed to have a solitary solution.

In case , but , we can use perturbation theory and look for approximate solutions of (5.1) which are close to solitary solutions of (5.2).

Using multiple time scales (a slow time and fast time scale such that ), we assume that the soliton solution can be expressed in the form [5, 15]
where , is a slow time, and is a fast time.

Substituting (5.3) into (5.1), then equating like powers of , we can obtain a series of linear equations for ), which can be solved sequentially.

In most cases, the nonlinear term in (5.1) plays an import role in understanding various solitary phenomena, and the coefficient is not limited to “small parameter.”

In order to overcome the shortcoming arising in the above solution process, hereby we applied the modified multitime expansions (see Section 2.9 of [16]). To illustrate the method, we consider the following equation:
Introducing the time scales , and using the parameter-expansion method (see Section 5.8), we assume that the solution and the constants and can be expressed as [5, 16]
where the constants and can be identified by means of no secular terms. Hereby, we define the secular term in a more general form that the term involves time in the form , even in case tends to zero when .

The equation for is
We can choose suitably the values of and , so that the solution of (5.8) can be easily obtained, and involves the basic properties of the original solution.

We use the Duffing equation to illustrate the solution procedure [16]
Suppose that the solution can be expressed in (5.5), and the coefficient, 1, can be expanded into
Substituting (5.5) and (5.10) into (5.9) and collecting terms of the same powers of , we have
where .

If only the first-order approximate solution is searched for, from (5.10), we have
or

The obtained frequency-amplitude relationship, (5.17), is valid for the whole solution domain, and the maximal relative error is less than 7% when .

5.3. Variational Approach

This section is an elementary introduction to the concepts of the calculus of variations and its applications to solitary solutions. Generally speaking, there exist two basic ways to describe a nonlinear problem: (1) by differential equations (DE) with initial/boundary conditions; (2) by variational principles (VP). The former is widely used, while the later is rarely used in solitary theory. The VP model has many advantages over its DE partner: simple and compact in form while comprehensive in content, encompassing implicitly almost all information characterizing the problem under consideration. Variational methods have been, and continue to be, popular tools for nonlinear problems. When contrasted with other approximate analytical methods, variational methods combine the following two advantages: (1) they provide physical insight into the nature of the solution of the problem; (2) the obtained solutions are the best among all the possible trial functions.

5.3.1. Inverse Problem of Calculus of Variations

The inverse problem of calculus of variations is to establish a variational formulation directly from governing equations and boundary/initial conditions. We will use the semi-inverse method [17, 18] to establish various variational principles directly from the governing equations.

Consider the well-known Korteweg-de Vries (KdV) equation
where and are constants, and the subscripts denote partial differentiations.

We rewrite it in a conserved form
According to the conservation form of (5.19), we can introduce a potential functional defined by
So the KdV equation can be written in the form
which can be derived from the following variational principle using the semi-inverse method [17]:
In order to obtain a generalized variational principle with two independent fields , we apply the Lagrange multiplier to (5.23)
The stationary condition with respect to results in
The Lagrange multiplier method is not valid for the case. This phenomenon is called Lagrange crisis. Hereby, we suggest three ways to overcome the crisis [18].

The Semi-Inverse Method [17, 18]Generally, the multiplier can be expressed in the form after identification
We replace the last term including the Lagrange multiplier by a new variable , that is,
where is unknown, which can be expressed in the form
Equation (5.27) is such constructed according to the semi-inverse method [17]. To identify , making the functional stationary with respect to , we have
where is the variational derivative defined as
Equation (5.29) should be equivalent to (5.20), and this requires
where is a nonzero constant. From (5.31), we can determine as follows:
where is a constant, . We, therefore, obtain the following needed variational principle:
Its Euler-Lagrange equations are
which satisfy the field equations (5.20) and (5.22), respectively.

The Hidden Lagrange Multiplier [18]Let us come back to (5.25), , which should be the constraint equation. This means that (5.25) inexplicitly involves a lost constraint equation, so we can identify the multiplier in the form [18]
This results in the same result above.

Replacement of Some Variables in the Original Functional Using the Constraint EquationSometimes the Lagrange crisis can be eliminated by replacing some variables in the original variational principle using the constraint equation; a detailed discussion was systematically given in [18]. We replace by in (5.23) and introduce a Lagrange multiplier in the resultant function
Identification of the multiplier yields
Submitting the identified multiplier into (5.36) results in
We find that the constraint is not eliminated yet, and this is another Lagrange crisis, which can also be eliminated by the semi-inverse method
The stationary conditions with respect to and are, respectively, as follows:
According to (5.20) and (5.21), we have
From (5.41), can be determined as
We, therefore, obtain the following variational principle:
By the semi-inverse method, we can obtain various different two-field variational principles, and we write here the following one for reference:
The potential in the above functionals (5.23), (5.33), and (5.44) requires second order of differentiation (), leading to the complications in the finite element calculation. For the purpose of simplification in finite element computation, we often introduce some additional variables to reduce the order of differentiations. This is of course equivalent to introducing some additional constraints in the variational principle. Generally, we can eliminate the introduced constraints by the Lagrange multiplier method, but as illustrated above, the method might fail.

Now we introduce a new variable defined as
By the semi-inverse method [17, 18], we obtain the following three-field variational principle:
It is obvious that all variables in the obtained functional (5.46) are in first-order differentiations, leading to much convenience in numerical simulation.

It is easy to establish a variational formulation by introducing a potential function, and we can also establish a variational principle without auxiliary special function. To elucidate this, we consider the KdV equation in the form
where subscripts denote partial differentiations. If we introduce a velocity potential defined as , then the KdV equation can be derived from the variational principle
where the Lagrangian can be expressed in the form
Our aim is to search for a Lagrangian for (5.47). It is easy to establish a variational formulation for differential equations with even orders. The KdV equation has odd-order differentiations, and therefore, no Lagrangian for (5.47). To circumvent this obstacle, we take partial differentiation with respect to to both sides of KdV equation, which turns out to be the following form:
By the semi-inverse method, we construct a trial Lagrangian in the form
where and are constants to be further determined. Its Euler equation can be readily obtained as follows:
or
Setting , then (5.53) turns out to be the modification version of the KdV equation (5.50).

Finally, we have the following needed Lagrangian in the form of velocity:
This approach can be extended to many nonlinear equations. Consider the modified KdV equation
Similarly, we change the equation so that it has even-order differentiations
By the same manipulation as illustrated above, we construct a trial Lagrange function in the form
where and are constants to be further determined. Its Euler equation can be readily obtained as follows:
or
or
Setting
(5.60) becomes (5.56), and we, therefore, obtain the following Lagrangian:
We can also use the semi-inverse method to establish a family of variational principles for a nonlinear system. We use one-dimensional traffic flow as an example.

The research on traffic flow began at the beginning of the 20th century. Lighthill and Whitham first proposed the fluid-dynamical model for traffic flow [19, 20]. The continuum equation for unsteady one-dimensional traffic flow can be, therefore, written as
where is the cross-sectional area of the road, is the velocity, is the density of cars, and is the source. The deficiency of the model is that the traffic flow actually cannot be considered as a continuum, and to eliminate this deficiency, a fractional differential model can be introduced:
where is the fractional differential, see Section 7.

In 1994, Zheng [21] suggested the following traffic flow model:
where is the possible maximal velocity, is a constant, and is the minimal traffic density when the cars can travel at a maximal velocity.

In order to establish a variational principle for the system, we rewrite (5.66) in the following equivalent form:
According to (5.65) and (5.67), we can introduce two functions and defined as
so that (5.65) and (5.66) are automatically satisfied.

The essence of the semi-inverse method [17, 18] is to construct an energy-like functional with a certain unknown function, which can be identified step by step. An energy-like trial functional for the discussed problem can be constructed in the following form:
where , , and are considered as independent variables, and is an unknown function of , and/or their derivatives.

There exist various approaches to the establishment of energy-like trial functionals for a physical problem, and illustrative examples can be found in [22, 23].

The advantage of the above trial functional lies on the fact that the stationary condition with respective to ,
leads to (5.67).

Calculating the functional (5.71) stationary with respect to and , we obtain the following Euler equations:
We search for such an , so that the above (5.72) satisfies the two-field equations. To this end, we set
From (5.73), we can immediately identify the unknown , which reads
So we obtain the following required variational functional:

Proof. The Euler equations of the above functional (5.75) are
From (5.68), we have . Substituting the result into (5.77) leads to . From the above three-field variational functional, we can easily obtain two-field or one-field variational function by substituting one- or two-field equations into the functional (5.75). For example, substituting into (5.75), we obtain a two-field variational functional
where the variable is not now an independent field. Further constraining the two-field functional (5.79) by the equation , we have
where is the traffic pressure defined as
The functional (5.81) has the same form of the well-known Bateman principle in fluid mechanics [18]. By a paralleling operation, we can also establish a variational functional with free fields , , and . A trial functional with an unknown function can be constructed as follows:
Here, the unknown is free from and its derivatives. By the same manipulation as illustrated above, we set
From (5.83), we can determine the unknown as follows:
So we obtain the following needed variational principle:
It is easy to prove that the Euler equations of the above functional (5.85) satisfy the field equations (5.67) and (5.69). Constraining the functional (5.85) by the equation , we obtain
which is under the constraint of the equation . Further substituting
into (5.86) results in
which is similar to the well-known Hamilton principle.

5.3.2. A Possible Connection between the Uncertain Principle and the Least Action Principle

Maupertuis-Lagrange’s principle of least kinetic potential action for a particle with mass can be expressed as follows [18]:
We rewrite (5.89) in the form
or
Equation (5.91) can be approximately written in the form
where . Equation (5.92) means that given that the particle begins at position at time and ends at position at time , the physical trajectory that connects these two endpoints is an extremum of , where is the standard deviation of the displacement, and is the deviation of the momentum.

For arbitrary or , the following inequality holds:
This is similar to the uncertainty principle.

In optics, Fermat’s principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light
Equation (5.94) can be approximately written in the form
where .

The light trajectory that connects these two endpoints and satisfies the following equation:
or
where , .

Consider a nonlinear oscillator with fractional potential [24]:
Its variational formulation can be readily obtained as follows:
Substituting (5.101) into (5.105), we obtain
Setting
we have
The exact frequency is . The 0.597% accuracy is remarkably good.

5.3.4. Variational Approach to Chemical Reactions

As an illustration, consider the following chemical reaction [25]:
which obeys the equation
where is the number of molecules at , is the number of molecules (or ) after time , and is a reaction constant. At the start of reaction (), there are no molecules (or ) yet formed, so that the initial condition is .

In order to obtain a variational model, we differentiate both sides of (5.110) with respect to time, resulting in
Substituting (5.111) into (5.110), we obtain the following second-order differential equation:
which admits a variational expression in the form
Its Hamiltonian, therefore, can be written in the form
where is a Hamiltonian constant, and it can be determined from initial conditions:
Equation (5.114) becomes
In view of the initial conditions and , (5.116) is equivalent to (5.110). This means that the variational principle, (5.113), is exactly equivalent to its differential partner, (5.110).

Assume that the solution can be expressed in the form
where is an unknown constant to be further determined.

Substituting (5.117) into (5.113), and setting , we obtain
So we obtain a first-order approximate solution for the discussed problem
In order to improve accuracy, we can assume that the solution can be expressed in a more general form
which should satisfy initial conditions , and this requires
Substituting (5.120) into (5.113), we set
Solving (5.121)-(5.122) simultaneously, we can easily determine parameters. The solution procedure is similar to that illustrated in [25], and we will not discuss in details to solve space.

We can also choose the following trial function:
where and are unknown constants to be further determined. It is obvious that (5.123) satisfies the conditions and . Submitting (5.123) into (5.113), and setting
we obtain with ease
Thus, we obtain the solution
which is the exact solution.

5.3.5. Variational Approach to Solitary Solution

In the review article [7], the variational approach to solitons was outlined by few lines, and now the method has been successfully applied to the search for soliton solutions [26, 27] without requirement of small parameter assumption, leading to an extremely simple and elementary but rigorous derivation of soliton solutions.

Considering the KdV equation, we seek its traveling wave solutions in the following frame:
where is angular frequency. Substituting (5.127) into (4.1) yields
where prime denotes the differential with respect to .

Integrating (5.128) yields the result
where is an integration constant, which can be determined from the initial condition. For solitary solutions or limit cycles, the solutions do not depend upon the initial condition, so we always set , and this results in
By the semi-inverse method [17], the following variational formulation is established:
Equation (5.130) also hints a Hamiltonian invariant, which reads
where is a constant, which can be determined by incorporating the initial condition. For solitary solutions or limit cycles, we set ,
Differentiating (5.131a) with respect to , we have
which is equivalent to (5.129b).

The Hamiltonian invariant (5.131a) and (5.131b) can also be used for construction of an asymptotic solution or used as an auxiliary function in the subequation method, see Section 5.10.

The semi-inverse method is a powerful mathematical tool to the search for variational formulae for real-life physical problems.

By Ritz method, we search for a solitary wave solution in the form
where and are constants to be further determined.